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Theorem oi3oa3 733
 Description: An attempt at the OA3 conjecture, which is true if (a ≡ b) = 1. (Contributed by Josiah Burroughs 27-May-04.)
Hypothesis
Ref Expression
oi3oa3.1 1 = (ba)
Assertion
Ref Expression
oi3oa3 (((a1 c) ∩ (b1 c)) ∪ ((((a1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) →1 c) ∩ (((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) →1 c))) = 1

Proof of Theorem oi3oa3
StepHypRef Expression
1 oi3oa3.1 . . . . . . . 8 1 = (ba)
21oi3oa3lem1 732 . . . . . . 7 (((a1 c) ∩ (b1 c)) ∪ (ab)) = 1
32lan 77 . . . . . 6 ((a1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) = ((a1 c) ∩ 1)
4 an1 106 . . . . . 6 ((a1 c) ∩ 1) = (a1 c)
53, 4ax-r2 36 . . . . 5 ((a1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) = (a1 c)
65ud1lem0b 256 . . . 4 (((a1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) →1 c) = ((a1 c) →1 c)
72lan 77 . . . . . 6 ((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) = ((b1 c) ∩ 1)
8 an1 106 . . . . . 6 ((b1 c) ∩ 1) = (b1 c)
97, 8ax-r2 36 . . . . 5 ((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) = (b1 c)
109ud1lem0b 256 . . . 4 (((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) →1 c) = ((b1 c) →1 c)
116, 102an 79 . . 3 ((((a1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) →1 c) ∩ (((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) →1 c)) = (((a1 c) →1 c) ∩ ((b1 c) →1 c))
1211lor 70 . 2 (((a1 c) ∩ (b1 c)) ∪ ((((a1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) →1 c) ∩ (((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) →1 c))) = (((a1 c) ∩ (b1 c)) ∪ (((a1 c) →1 c) ∩ ((b1 c) →1 c)))
13 ax-a2 31 . 2 (((a1 c) ∩ (b1 c)) ∪ (((a1 c) →1 c) ∩ ((b1 c) →1 c))) = ((((a1 c) →1 c) ∩ ((b1 c) →1 c)) ∪ ((a1 c) ∩ (b1 c)))
141r3a 440 . . . . 5 b = a
1514ud1lem0b 256 . . . 4 (b1 c) = (a1 c)
16151bi 119 . . 3 1 = ((b1 c) ≡ (a1 c))
1716oi3oa3lem1 732 . 2 ((((a1 c) →1 c) ∩ ((b1 c) →1 c)) ∪ ((a1 c) ∩ (b1 c))) = 1
1812, 13, 173tr 65 1 (((a1 c) ∩ (b1 c)) ∪ ((((a1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) →1 c) ∩ (((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (ab))) →1 c))) = 1
 Colors of variables: term Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44 This theorem is referenced by: (None)
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